Atlas › Modular forms › Weight 2 › Level 158 › 158.2.a

Newspace 158.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(158))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
158.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$	158b1 rank 1
158.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 3q^{5} - q^{6} - q^{7} + O(q^{8})$	+ −	$+1$	158d1 rank 0
158.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} - 3q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$	158a1 rank 1
158.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$	158c1 rank 0
158.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} - 2q^{5} + 2q^{6} + O(q^{8})$	− +	$+1$	158e1 rank 0
158.2.a.f	2	$\mathbb{Q}(\sqrt{6})$	$2q - 2q^{2} + 2q^{4} - 4q^{5} + 8q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
158.2.a.a	$x - 9$
158.2.a.b	$x - 5$
158.2.a.c	$x + 41$
158.2.a.d	$x - 31$
158.2.a.e	$x + 32$
158.2.a.f	$x^{2} - 28x + 190$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(158))$	7
Old forms	12
Cusp forms $S_{2}(\Gamma_0(158))$	19
Eisenstein series $E_{2}(\Gamma_0(158))$	3
Modular forms $M_{2}(\Gamma_0(158))$	22

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(158)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.307 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 2, 79, 158]: [0, 1, 7, 22] (certified at ell = 2147483053)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(158, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 158:2 1000

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